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Second, we revisit the transient behavior in the case of a constant reset rate and a constant or linear growth rate, improving on a previous analysis by including more general initial conditions.<\/jats:p>","DOI":"10.3390\/math10040644","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T08:22:19Z","timestamp":1645431739000},"page":"644","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Comments on Mathematical Aspects of the Bir\u00f3\u2013N\u00e9da Model"],"prefix":"10.3390","volume":"10","author":[{"given":"Ilda","family":"In\u00e1cio","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, 6201-001 Covilh\u00e3, Portugal"}]},{"given":"Jos\u00e9","family":"Velhinho","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias and FibEnTech-UBI, Universidade da Beira Interior, R. Marqu\u00eas D\u2019\u00c1vila e Bolama, 6201-001 Covilh\u00e3, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"032130","DOI":"10.1103\/PhysRevE.95.032130","article-title":"Dynamical stationarity as a result of sustained random growth","volume":"95","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.physa.2018.02.078","article-title":"Unidirectional random growth with resetting","volume":"499","year":"2018","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"e0179656","DOI":"10.1371\/journal.pone.0179656","article-title":"Science and Facebook: The same popularity law!","volume":"12","author":"Varga","year":"2017","journal-title":"PLoS ONE"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bir\u00f3, T.S., N\u00e9da, Z., and Telcs, A. (2019). Entropic Divergence and Entropy Related to Nonlinear Master Equations. Entropy, 21.","DOI":"10.3390\/e21100993"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"124491","DOI":"10.1016\/j.physa.2020.124491","article-title":"Scaling in income inequalities and its dynamical origin","volume":"549","author":"Gere","year":"2020","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"126194","DOI":"10.1016\/j.physa.2021.126194","article-title":"Wealth distribution in modern societies: Collected data and a master equation approach","volume":"581","author":"Gere","year":"2021","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"23","DOI":"10.3389\/fphy.2022.827143","article-title":"Wealth distribution in villages. Transition from socialism to capitalism in view of exhaustive wealth data and a master equation approach","volume":"10","author":"Gere","year":"2022","journal-title":"Front. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"292","DOI":"10.3938\/NPSM.70.292","article-title":"Wealth Distribution for the Spin Agent Model of the Stock Market","volume":"70","author":"Park","year":"2020","journal-title":"NPSM"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"124201","DOI":"10.1016\/j.physa.2020.124201","article-title":"Wealth distribution models with regulations: Dynamics and equilibria","volume":"551","author":"Cardoso","year":"2020","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"125283","DOI":"10.1016\/j.physa.2020.125283","article-title":"A simple and efficient kinetic model for wealth distribution with saving propensity effect: Based on lattice gas automaton","volume":"561","author":"Cui","year":"2021","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_11","unstructured":"Van Kampen, N.G. (1992). Stochastic Processes in Physics and Chemistry, Elsevier Science B.V."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ross, S.M. (2019). Introduction to Probability Models, Academic Press.","DOI":"10.1016\/B978-0-12-814346-9.00006-8"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"205","DOI":"10.2307\/2343842","article-title":"The generalized Waring distribution applied to accident theory","volume":"131","author":"Irwin","year":"1968","journal-title":"J. Roy. Stat. Soc. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"10837","DOI":"10.1038\/s41598-018-28962-1","article-title":"How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions","volume":"8","author":"Hanel","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bir\u00f3, T.S., Csillag, L., and N\u00e9da, Z. (2021). Transient dynamics in the random growth and reset model. 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An Introduction to Probability Theory and Its Applications, John Wiley nad Sons, Inc."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/4\/644\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:22:55Z","timestamp":1760134975000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/4\/644"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,19]]},"references-count":18,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["math10040644"],"URL":"https:\/\/doi.org\/10.3390\/math10040644","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2022,2,19]]}}}